Griffiths Introductory to Quantum Mechanics (3rd Edition):
Problem 7.9
Problem 7.9 Consider a particle of mass...
Problem 7.9 Consider a particle of mass m that is free to move in a one- dimensional region of length L that closes on itself (for instance, a bea that slides frictionlessly on a circular wire of circumference L, as Problem 2.46) (a) Show that the stationary states can be written in the form 2π inx/L where n 0, t l, 2, , and the allowed energies are In Notice that-with the exception of the ground state (n = 0)-these are all doubly degenerate. (b) Now suppose we introduce the perturbatiorn where a 《 L. (This puts a little "dimple" in the potential at x = 0, as though we bent the wire slightly to make a "trap".) Find the first-order correction to En, using Equation 33. Hint: To evaluate the integrals, exploit the fact that a«L to extend the limits from L/2 too0; after all, H' is essentially zero outside (c) What are the "good" linear combinations of Vn and ψ-, for this problem? (Hint: use Eq. 7.27) Show that with these states you get the first-order correction using Equation 7-9. (d) Find a hermitian operator A that fits the requirements of the theorem, and show that the simultaneous eigenstates of HO and A are precisely the ones you used in (c).